The scores of all applicants taking an aptitude test required by a law school have a normal distribution with a mean of 420 and a standard deviation of 100. A random sample of 25 scores is taken.
a. Find the probability that the sample mean score is higher than 450.
b. Find the probability that the sample mean score is between 400 and 450.
c. The probability is 0.10 that the sample mean score is higher than what number?
d. The probability is 0.10 that the sample mean score is lower than what number?
e. The probability is 0.05 that the sample standard deviation of the scores is higher than what number?
f. The probability is 0.05 that the sample standard deviation of the scores is lower than what number?
g. If a sample of 50 test scores had been taken, would the probability of a sample mean score higher than 450 be smaller than, larger than, or the same as the correct answer to part (a)? It is not necessary to do the detailed calculations here. Sketch a graph to illustrate your reasoning.